TITLE: Any ball is connected?
QUESTION [6 upvotes]: Let $X$ be a compact , metric space. 
Assume that the closure of every each open ball it the closed ball with same center and radius. 
Prove that any ball in this space is connected.

REPLY [2 votes]: Assume that $X$ is not connected. Then there exists a set $F\subset X$, such that $F, K=X\smallsetminus F$ are non-empty, open and closed, and as they are closed they are compact.
This means that 
$$
r=\mathrm{dist}(F,K)>0,
$$
and further there exist $x\in F$ and $y\in K$, such that $d(x,y)=r$. 
In particular, $$\big(F\cup B(x,r)\big)\cap K=\varnothing,$$
and thus
$$
F\subset F\cup B(x,r)\subset X\smallsetminus K=F,
$$
which implies that $F= F\cup B(x,r)$ and
$$
F=\overline F= \overline{F\cup B(x,r)}=\overline F\cup \overline B(x,r),
$$
which is a contradiction since $y\not\in F$, but $y\in \overline B(x,r)$.